Continuous function between a lower semi-continuous function and an upper semi-continuous function.

Question

Let $X$ be a compact metric space, $u: X \to [0, 1]$ an upper semi-continuous function and $l: X \to [0, 1]$ a lower semi-continuous function such that $u(x) < l(x)$ for each $x \in X$.

Does there exist a continuous function $f: X \to [0, 1]$ such that $u(x) < f(x) < l(x)$ for each $x \in X$?

Answer 1

It is true. See the book

Engelking, Ryszard. "General topology."

On p.428 5.5.20 you find the following result as an exercise:

A $T_1$-space $X$ is normal and countably paracompact if and only if for each pair $f,g$ of real-valued functions on $X$, where $f$ is upper semicontinuous and $g$ is lower semicontinuous such that $f(x) < g(x)$ for all $x \in X$, there exists a continuous $h : X \to \mathbb R$ such that $f(x) < h(x) < g(x)$ for all $x$.

You will also find references to papers containing proofs, for example

Miroslav Katětov, On real-valued functions in topological spaces, Fundamenta Mathematicae 38 (1951), 85–91

as quoted in Dave L. Renfro's link.

Answer 2

Yes. See Katětov–Tong insertion theorem. I suspect this result for your context (compact metric space domain) was proved earlier, but I off-hand I don't know. FYI, googling semicontinuous + insertion + theorem will lead to many similar results. (a few minutes later) I just noticed you have strict inequality, whereas the theorem I cite involves non-strict inequality. I'm not sure whether your version can be obtained from what I gave, at least unless we assume (everywhere locally) a positive lower bound on the pointwise differences between the semicontinuous functions.